Dept. for Speech, Music and Hearing

Quarterly Progress andStatus Report

Parameters influencingeigenmodes of violin plates

Molin, N-E. and Lindgren, L-E. and Jansson,E. V.

journal: STL-QPSRvolume: 27number: 1year: 1986pages: 111-138

http://www.speech.kth.se/qpsr

IV. MUSIC ACOUSTICS

A. PAFWEIERS 1-ING EIGHYPKlDES OF VIOLIN PLATES

Nils-mik Molin*, Lars-Erik Lindgren*, and Erik Jansson

Abstract Assuming orthotropic behavior, material properties are easily de-

termined for violin blanks made of spruce and of maple. Theoretical calculations, by a computer program for the finite element method (FEM), are compared with experiments. An FFrmodel of a violin plate, symmetri- cal along the center line (the glue joint), is used when calculating the eigenmodes. Shape and frequency of the first six modes are studied when 1) plate arching, 2) plate thickness and thickness distribution, 3) material parameters for spruce and maple, and 4) boundary conditions of the plate are used as parameters. Results are presented in figures and tables.

For comparison a violin top and a violin back plate are made of blanks with the material properties as and measures close to those of the FE-model. Calculated and experimentally determined modal shapes and frequencies are found to be close - the frequencies differ often much less than 8%. Thus, we believe that the calculated dependence on prop- erties on different parameters of the violin plate can be trusted.

Pules to adjust thickness and arching are discussed starting from material properties of the blanks. Such information is needed by the violin maker aiming at specific frequencies for certain eigenmodes for the free-free plate. It is also shown that changes in boundary condi- tions affect modes of vibration strongly.

1. Introduction The violin maker selects his wood material and works his plates to

specific arching and thickness. Thereby, he can monitor the work by the changes in eigentones as the work proceeds. He can also use Chladni patterns. Typical patterns of the free-free plate eigenmodes can be found, and rules on how to ad just, "to tune", the plates exist (Hut- chins, 1980; 1985). Finally, the different parts are assembled into a violin.

In this adjusting process the maker tries to give the free-free plate specific properties wanted. The reason why the properties should be such, or how much additional information is needed to describe the properties of the assembled instrument is still to be found. As a matter of fact the existing rules for adjustments need verification. Therefore, we will in this work investigate the influence of thickness and thickness distribution, arching, and material properties. In addi- tion, we will start to investigate the influence of different boundary conditions, a parameter that strongly influences the final result.

* ~ l & University of Technology, L,ul&, Sweden

As a starting pint we used the material properties determined by means of resonance frequencies of the blanks (Molin, Tinnsten, Wiklund, & Jansson, 1984). Numerical models, i.e., FE-models of the violin plates, were used. Resonance frequencies and eigenmodes were calculated. Tb check the calculations, a top and a back plate were made from the blanks to the measures of one of the FE-models. As the material proper- ties used in the calculations and the physical plates were the same,

direct comparisons to evaluate the accuracy of the calculations were possible.

The results of our calculations were compared with the results of the experiments, i.e., frequencies and shapes of vibration modes of the blanks and those of the finished violin plates. Thereby, we found good agreement between the calculated and the experimentally determined pro- perties. The g c d agreement is somewhat astonishing since there are so many reasons why they could differ; real wood has deviations from an assumed symmetry and homogenity, making a violin plate identical to that of the F&model is not possible, wood changes properties with age and humidity, there are experimental errors, numerical errors, etc.

By the numerical and physical experiments we wanted to answer the following questions:

I. I-bw do changes in thickness, thickness distribution, arching, ad stiffness distribution influence the eigenmodes of the free violin plate?

11. Can "deviating" material properties be "corrected" for by ad- justing thickness and arch height?

111. If so, how should the thickness and arching be chosen with given material properties of the blatiks (i.e., mass and resonance frequencies of the blank)?

N. What kind of boundary condition is appropriate for the plate when glued to the ribs?

2. Starting pint - measured and calculated properties of the blank Several parameters are important for the vibration behavior of wood

in string instruments. Nine elastic parameters are needed to describe an arched shell of wood, when it is assumed that the plate is cut f r o m nonsymmetrical samples of wood (neither in parallel with nor in per- pendicular to the grain of the wood, McIntyre & Woodhouse, 1984; 1985). ?he parameters are likely to be dependent on frequency and amplitude of vibration (nonlinear). Furthermore, they can be sensitive to aging and internal damping of the wood.

In this work we instead assumed that wood can be described as a perfectly quarter-cut thin plate of wood, thus, we needed only four

elastic constants for the description. Furthemre, we assumed that the parameters are independent of frequency, internal damping, humidity, aging etc. In such a simple model one cannot expect that every aspect of the vibration properties can be analyzed, still this model is con- siderably more efficient than using two one-dimensional beams cut alcng and across the grain, respectively.

Thus, it was assumed that the violin plates could be described as a thin and slightly curved shell of an orthotropic linearly elastic mate- rial. The material properties needed are the density of the wood, the two moduli of elasticity EL along and E;r across the grain, respective- ly, the in-plane shear modulus G (twisting modulus), and one of the Poisson contraction numbers (vm or vLT). The second Poisson number is related to the first by the equation

The number vm is the ratio between the relative compression in the G direction and the relative elongation in the T-direction when the sample is stretched in the %direction. The number Vm is shortly referred to as v in this report.

The blanks have a "standard" size of 385 x 215 x (7.5 to 20) mm3, where the 20 mrn is the thickness at the glue joint and 7.5 mm is the thickness along the edges in parallel with the glue joint. Qle side of the blank is flat so the blank is in fact a slightly curved shell, see Fig. 1.

The "standard" FE-model consisted of 48 triangular three-node flat shell elements arranged to simulate a quarter of the symmetric violin blanks. With combinations of symmetric and/or antisymmetric boundary conditions, it was possible to simulate the vibrations of a free-free plate. In addition, calculations were made with twice as many elements and on free-free plates to assure that the discretization error was small, at least for the three lowest eigenmodes. Material properties determined from these blanks were later on used in the calculations on the arched violin plates.

2.1. Spruce blank Physical experiments with the spruce blank gave the resonance

frequencies 272, 562, and 631 Hz for the 1/11 2/0, and 0/2 modes, respectively (1/2 stands for one nodal line across the grain and two nodal lines along the grain). The geometry of the blank was 386 x 215 x (20 - 8.5) mm3, which was somewhat different from our "standard size". The material properties (except the density) were adjusted so that the calculated eigenmodes for the blank agreed well with the measured eigen- modes. The material parameters calculated for the blank were closely the same as those obtained from physical testbars. m e differences were in general considerably less than 10%. The values for the blank are density, EL, %, GI v M 2 kg/m3, 16 GPa, 0.975 GPa, 0.725 GPa, and 0.03,

I I

t h h f h

Fig. 1 . Geanetry and measures of "standard" blanks, L = 385 m, B = 215 m, bln = 20 mn, and hh = 7.5 m (Mattsson, 1 984) .

Fig. 2. a) Calculated iso-amplitude lines and measured nodal lines for the spruce blank.

b) Calculated iso-amplitude lines and measured nodal lines for the maple blank. (Iso-amplitude lines and glue joint are marked with thin lines, calculated nodal lines w i t h thick lines, and measured nodal lines with broken lines.)

Fig. 3. F E d e l of one half of the violin plate.

reason £or making a symmetrical model was economical. It was cheaper to make two calculations with symmetrical and antisymmetrical boundary conditions along the joint than one calculation on a complete model.

3.1. Nmerical FEEI-experiments for spruce The same material parameters as for the spruce blank was used,

i.e., density EL, GI ~:482 kg/m3, 16 GPa, 0.975 GPa, 0.725 GPa, ard 0.03, respectively. In all tables, 13.5/3.5 for instance, means a maximum arch height of 13.5 mm and a constant thickness of 3.5 mm. The headings "sym" and "asym" stand for symmetrical and antisymmetrical modes, respectively. The numbers given are frequencies in Hz for the first six d e s .

Table I. Resollance frequencies in Hz for variations of arch height and thickness, spruce.

Correspnding modal shapes for 13.5/3.5 are given in Fig. 4a.

Table 11. Resonance frequencies in Hz for variations of thicknesses in different areas, spruce, (cf., Fig. 6).

+ area 4

13.5/3.5 area 1 + area 2 + area 3 = 13.5/2.5

sym asym sym asym sym asp sym asym sym asY='

Table 111. Resonance frequencies in Hz for variations of material param- eters (one at a time) at constant arch height/thickness 1612.5, spruce.

Fig. 4. a) FEM-calculated mdal shags for spruce plate 13.5/3.5 (nmbers mark frequencies in Hz, thin lines are iso- amplitude lines and thick lines are nodal lines, plus and minus signs mark antinodes of opposite vibration phrases) .

b) Experimentally determined nodal lines for modes no 1 to no 5. (Left figure: mode no 1 at 105,Hz bmken dotted line, mde no 2 at 200 &,broken line, mode no 5 at 448 Hz, full line. Right figure mode no 3 at 332 Hz, broken lines and shadawed area, and mode no 4 at 317 Hz, full lines. )

SZL

3.2. Numerical FEM-experiments for maple The same mater ia l parameters a s fo r t he maple blank were used,

i.e., density, EL, GI v :609 kg/m3, 12 GPa, 1.96 GPa, 1.725 GEa, and 0.02, respectively.

Table IV. Resonance frequencies i n Hz for variations in arch height.

-1 shapes for 13.5/3.5, maple are given i n Fig. 5a.

4. Free-free plates - &mica1 exrx?riments and comparisons with FEM-experiments

4.1 Final shapes of plates The top plate was worked down in ten steps, and in each step the

frequencies and the nodal patterns of the f i r s t , second and f i f t h modes were measured. After the l a s t s t ep the t h i rd and fourth modes w e r e measured too. In i ts f i n a l shape i ts measures w e r e 13.0/3.5 with an accuracy of i-0.2 mm. The back plate w a s worked down in nine steps to measures 16.5/3.5 +- 0.2 mm. In i t i a l diff icult ies, however, in machining enforced a thickness of 2.5 mm along the edges i n a band i n parallel with the inner contour of the ribs (this geometrical error should give opposite effect with glued edges). The accuracy of frequency measure- ments were i-2 Hz.

Table V. Masses and frequencies i n f inal stages.

Resonance frequency (EW) for nude Mass(g) n o 1 m 2 n o 3 no4 n o 5

a m sym sym asym sym

Fig. 6. Definitions of areas 1 to 4 used in the stepwise continued thinnings.

can be found at nodal crosses, where the phase has equal values across the diagonal of the cross.

Since we assume that the kinetic and elastic energies interchaqe the same amount of energy (we also assume that the damping and the driving forces are constant and small), small perturbations of the state of dynamic equilibrium will result in a change of the resonance frequen- cy because it is only the kinetic energy that depends on frequency. If the elastic energy is increased but the parameters contributing to the kinetic energy are kept constant, this will result in an increase of the resonance frequency (and vice versa).

If the density of the plate is increased this will only contribute to the kinetic energy and thus lower the resonance frequency. An in- crease in the values of elastic parameters will contribute to the elas- tic energy and thus increase the resonance frequency.

An overall thinning of a free-free plate lowers all frequencies, i.e., an overall decrease in thickness affects the "stiffness more than the mass", although both elastic energy and kinetic energy are de- creased. If the plate, however, is thinned in a smaller area in a place where the plate is moving more like a rigid body, that is neither bent or twisted, it can be expected that the corresponding resonance frequen- cy will be raised. The vibrating mass and the inertia are in this case decreased in a position where its bending or torsional stiffness are of minor importance. Areas of a plate that moves in such a way can be found in the EEM-calculated modes of vibrations in the same way as in time-average inter ferograms i .e., areas where the iso-ampli tude contour lines are at equal distances across the plate. On the other hand, if thickness of the plate is removed in areas where the iso-amplitude lines show a strong bending or twisting, i.e., the neighboring iso-amplitude lines are far from equi-distant, it can be predicted that the resonance frequency of that mode will be lowered. With these rules in mid and with the iso-amplitude curves at hand, one should at least in principle be able to predict how a thinning of the plate affect a specific reso- nance frequency. Areas where the plate moves like a rigid body are often found close to the edges of free plates or close to nodal lines where the amplitude of vibration shifts sign when crossing the mdal line, i.e., the body is tilting. If two nodal lines are crossing, however, the amplitude of vibration has equal signs across the diagonal of the nodal cross, indicating that the plate is twisted. In this case, it can be predicted that the frequency will probablybe lowered when the plate is thinned in the area of the crossing. Strong bending is usually found in the middle between the nodal lines, cf., the sinusoidal curve.

In this way it, is possible to predict in a qualitative way what will happen when small changes are made but as quantitative results are needed, results from FEM-calculations are presented in next section.

5.2. Quantitative results of nmerical perturbations

This section is based on data in Section 3. ?he top plate was made of spruce with nominal material parameters, i.e., density, EL, %, GI

"7482 kg/rn3, 16 GPa, 0.975 GPa, 0.725 GPa, and 0.03, respectively. It had a nominal (arch height/thickness) 16/2.5 with free-free boundary cadi- tions .

W l e VI. Relative frequency shifts in 8 for 10% shifts in parameters.

Changes in mode n o 1 n o 2 n o 3 n o 4 n o 5 no6 asym sym sym asym sym asym

a) density +lo% -5.0 -5.0 -5.0 -5.0 -5.0 -5.0

E~ +lo% 2.2 1.2 2.1 1.6 2.9 1.4

% +lo% 1.3 1.7 1-0 1.1 1.7 1.1 G +lo% 3.1 0.1 0.9 1.3 2-1 1.3

b) thickness +lo% 6.8 7.3 7.0 5.8 5.5 6.5

c) - . . 53 -lo% (16/3.5to 16/3.05, cf., Fig. 6)

in area 1 + area 2 + " 3 + " 4

results relative former rww

d) arch height +lo% (fran 13.5/2.5 to 14.85/2.5)

2 5 2.4 2.4 3.5 2.8 3.0

5.3 Mean influence on all modes

Table VII. Geometrical parameters, material properties, and mean fre- quency shifts of the first six modes.

10% increase in overall thickness gives +6.5% in mean frequency shift I# density -5% I1

II arch height +2.8 I#

The three material parameters give a mean influence of 4.7% if all three are increased 10% at the same time. The influence of a 10% change

in the Poisson number is negligible. It should be noted that a 6% change in frequency corresponds to a semi-tone step.

Conclusion: Changes in density, overall thickness, and arch height affect all first six modes in a more or less equal way, as can be seen in Table VII. In affecting the average resonance frequencies of the first six modes a +lo% overall thickness increase approximately equals -12% density change, or 20% of arch height change, or +12% increase in all the three elastical parameters. These figures can be used as guide lines to correct for deviating initial parameters.

5.4. Influence on single modes Thinning the plate at different positions affects modes differ-

ently, from -4.7% to +1.4% as given in Table VI. Thus, modes of vibra- tion can be "tuned" by thinning the plate at specific positions.

This is possible since by thinning a plate in an area, mass and inertia are removed at the same time as bending and torsional stiffness are decreased. As these simultaneous operations affect the resonance frequency of a specific mode in different directions, resonance frequen- cies can be tuned up or down depending on which factor is the stronger one. As it is interesting to see both how a single mode is changed by different properties and how a single property affects all modes, the analysis of results is presented below in both ways.

5.4.a. Modes no I, no 2, and no 5, "the mst important modes" MODE NO 1:

This mode is asymmetric relative the center line (the glue joint), i.e., the center line is a nodal line (longitudinal), and the vibrations are of opposite phases on each side of the center line, cf., Figs. 4 and 5. It has a second nodal line (transversal) cutting the first nodal line in perpendicular and, approximatively, in the middle of the plate. In the four "corners" (the plate is here regarded as a rectangular plate

with four "corners", not meaning the violin corners at the C-bouts) there are four antinodes of about equal amplitude. This type of vibra- tion is a twisting mode around the nodal cross, as the phase across the diagonal is the same.

Reducing thickness in the area around the Modal cross (area 4 in Fig. 6) gives a large influence on the resonance frequency and reducing thickness in areas along the edges (areas 1 and 2) gives a small influ- ence. The shearing modulus G is the material property which influences the resonance frequency the most.

MDDENO 2: The second mode is symmetrical (cf., Figs. 4 and 5 ) , which means

that the derivative of the vibration amplitude is zero across the center line altlmugh fhe vibration amplitude is not zero. This mode has two (bent) nodal lines mainly in parallel with the symmetry line. An in- vestigation of the variation in distance between the iso-amplitude lines shows that the plate at the same time is bending both longitudinally and tranmersally. The mode has four antinodes at the edges with maxima at the C-bouts (in the middle of the "longer sides").

A thinning at areas 2, 3, and 4 gives a large influence on the resonance frequency, especially thinning in area 4. A thinning along the edges raises the frequency. The influence of the modulus of the elasticities EL and % is fairly large, much larger than the influence of the shearing modulus.

MODE NO 5, THE RING MODE:

This famuos symmetrical mode is generally referred to as the ring mode, as the internal antinode is enclosed or close to be enclosed by a pear-shaped nodal line, cf ., Figs. 4 and 5. The edges of the plate a d the antinode have approximatively the same amplitudes and vibrate in opposite phases. The mode can also be regarded as two cooperating modes, one with transversal nodal lines and a second one with longitudi- nal nodal lines, which are vibrating say 90 degrees out of phase forming a closed nodal line, i.e., the plate is thus at the same time bending both longitudinally and transversally and twisting with moving "nodal corners". This mode can easily be detected by the ear by holding the plate at a nodal line and tapping the plate in the center.

The resonance frequency is influenced the most by thinnings in areas 2 and 3 and the least for thinning along the edges. All material properties influence this mode by approximately an equal amount.

5.4.b. Mdes no 3, no 4 and no 6 MODE NO 3 :

Mode no 3 is symmetrical with three bent nodal lines, one in the upper part mainly transversal, and two in the lower part mainly longi- tudinal, cf., Figs. 4 and 5. This mode is often close in frequency to the next one and these two modes are sometimes shifted in frequency

order. The plate is vibrating mainly in its outer parts and with a relatively small vibration amplitude in the center part of the plate. When excited with small vibration amplitudes, it looks as if the major part of the middle is a "nodal area" with the shape of a six-pointed star. The plate is mainly bending longitudinally in its upper part and both transversally and longitudinally in its lower part.

The resonance frequency is much influenced by thinnings in areas 2 and 3 but little in area 1. A change in longitudinal modulus of the elasticity EL gives a larger influence than that of EL, or G.

This mode is asymmetrical, cf., Figs. 4 and 5 . The mode is, as previously said, often interchanged in frequency order with the pre- vious mode. Mode no 4 shows similarities with the first mode, but it has two transversal nodal lines, one in the u p p r part and one in the lower part, both cutting the longitudinal nodal line. Sometimes these transversal nodal lines meet in the center of the plate which gives a pattern like a six-pnted star but this time with a nodal line along the glue joint. Along the edges there are six antinodes with approximately the same amplitude. Tkis mode of vibration can be described as a twist- ing of the plate along the symmetry line at the two m a 1 crosses, and, in addition, it has some bending.

This mode is the least effected by thinning in area 1, i.e., along the edge, and depends equally on EL, % and G, approximately, that is, the same as for mode no 5.

MODE NO 6: The sixth mode is asymmetrical with three transversal nodal lines

mainly across the plate in perpendicular to the nodal line along the glue joint. Sometimes two of the transversal nodal lines degenerate into lines more parallel to the glue joint in the lower part of the plate, not crossing the symmetry line, as in Figs. 4 and 5. The plate vibrates with eight antinodes at the edges often with low amplitude in the center of the plate. The plate is twisting as well as bending transversally and longitudinally.

The resonance frequency is most influenced by thinnings in areas 2 and 3 and the least by thinnings at the edges in the center. ?he influence of the material properties is about the sane as for mode no 4, another asymmetrical mode.

5.5. Influence by single parameters In this part we are only investigating the "most important" modes,

i-e., modes no 1, no 2, and no 5.

Canparisons of the results from thinning of the plate: Thinning all of the plates gave approximately the same influence

for mode no 1 and no 2 but somewhat less for mode no 5 (cf., Table

VI.b). Thinnings of the physical spruce and maple plates (considerably thicker than the FEM-plates) all over gave approximately double as large frequency decreases for modes no 1 and no 2 as those for mode no 5. The deviations between experimental and calculated results decreased with thinner physical plates. Thus, the experiments and the calculations are in large in agreement.

Thinning along the edge (area 1) increases mode no 2 much more than the athers are changed; thinning in area 2 decreases mode no 1 much less than the others; thinning in area 3 decreases mode no 2 somewhat more than no 1 and no 5; thinning in area 4 decreases modes no 1 and no 2 much more than mode no 5.

The results calculated for area 4 were clearly verified by the thinnings of the physical spruce and maple plates. Tbse for area 2 were consistent but less clear. m r area 3 the experiments gave larger in- fluence on modes no 1 and no 2 than on mode no 5 which is partially contradictory to the calculations. Thus the experimental and calculated results are in large in agreement again.

Examples of how thinning influences the shape of mode no 5 are given in Fig. 7b. We can here see that the lower left "corner" of the nodal lines is shifting towards the edge and that the position of the antinode at the lower left bout is shifted upwards. This result is typical for the nodal lines, when the plate is made thinner. The nodal

lines become more rectangular in shape and the lower corner tends "to spill wer" the edges. The same tendency is clear from thinnings of tlle physical plates. The antinode in the center seems not to be shifted.

Examples of results of changing the material parameters: Changes in the elastical material parameters also change different

modes differently. The changes of resonance frequencies are from -0.2% v to +3.1% (G) for 10% shift in the material parameters.

The influence on the shapes of mode no 5 is shown in Fig. 7a. We can here see that a 10% increase in the EL results in that the lower "corner" of the nodal line "spills over the edge" but no other clear difference - the antinodes do not shift positions. A 10% increase for 5 and G gives negligible influence. In addition, the numerical experi- ments showed that the spruce plates had somewhat more pronounced "cor- ners" than the maple plates of the same dimensions.

An increase of the Poisson ratio with 100% (from 0.03 to 0.06) increases the frequency of modes no 1 and no 5, i.e., the first torsional mode and the ring mode approximately 2.7%, but the other resonance frequencies decrease approximately 1%. The shifts in frequency and in the nodal lines are the same as for EL +lo%, cf ., Fig. 7a. Examples of the results of differences in arch height

Increased arch height (13.5 to 16) gives a 5% increase in resonance frequencies. There is a tendency for a larger shift for mode no 5

Fig. 7. a) Variations in shape of mode rn 5 for different %,$, Gt and v f ran FEM-calcul lo s.

Fig. 7. b) Variations in shape of mode no 5 for successive thinnings in dif- ferent areas f m FFM-calculations.

(clear for maple) than for mode no 1 and no 2. The nodal lines of mode no 5 become more "rectangular" with increased arch height. The position of the central antincde did not shift but that of the lower bout was somewhat shifted (at the lower left edge, cf., Fig. 7b).

If the elastical parameters are known, certain corrections for deviations in stiffness are sometimes possible to correct for by changes in geometry. A high value for the XI for instance, is counteracted by decreased arch height.

5.6. A practical example Suppose you have decided to make a violin top plate with a geometry

close to that of our plate (an approximation of a Stradivarius plate). You have carefully chosen a piece of spruce and machined it into a blank of the dimensions 385 x 215 x (20 to 7.5) mm3, cf., Fig. 1. You measure its density (mass) and the frequencies of the first three modes of vibration by tapping or by a Chladni method. You will probably find the modal shapes to be close to those in Fig. 2a. If the plates have large deviatons in symmetry, this will be seen in the modal patterns and you are likely to be better off by starting with a new blank. If the modal patterns are right, then you compare your results with the density (the mass), the frequencies of modes no 1, no 2 and no 3, i.e., 482 kg/m3, 272 J32, 560 HZ, and 630 HZ, respectively. Calculate the relative dif- ference in % from our results.

It is reasonable to work with the assumptions (Molin & al., 1984) that:

rode no 1 depends only on the G-modulus

This means that a +5% deviation in frequency for a mode gives a +lo% deviation in the corresponding material parameter. This fact is hardy to use for inter- and extrapolations.

A +lo% deviation in density (mass) will shift all frequencies -58, and you should correct for this first. Thereafter, you can calculate an estimate of your material parameters from the presented parameters, i.e., EL, ETl and G (16 GPa, 0.975 GPa, and 0.725 GPa). You will now have your own set of material parameters for density, EL, E+r and G.

The next step will be your choice of suitable resonance frequencies for modes no 1, no 2, and no 5 of your free-free violin plate (cf., Fig. 4)

In Table VI and the following discussion you will find the influ- ence of your deviations in parameters from our results for a 16/2.5 mm

violin top plate, that have frequencies of 88 HZ, 157 Hz, and 416 Hz for these three modes. This will enable you to choose a suitable arch height /thickness of your own. After you have made this plate you should meas- ure the resonance frequencies for your modes. !they shold hopefully be a bit to high, and p u should adjust them by thinning the plate in a m r e I

priate areas, also given in Table VI.c. The outlined procedures works also for the back plate.

6. Fastened edcres: FEM-experiments and physical experiments

6.1. Top plate, experimental results The v i o l i n p l a t e (13.0/3.5) was glued t o t he ribs of t he v i o l i n

(with paper i n the glue joint to make it easy and safe to take apart). The other side of the ribs w e r e glued to a heavy fixture w i t h a large hole on the back side. mis arrangement allowed u s to use the Vib ra -

vision (an electronic speckle pattern interferameter using a laser and presenting the vibration patterns on a TV-monitor) to determine shapes and frequencies of the eigenmodes (Ek & Jansson 1984). Thereby, the vibrations (now too s m a l l for the Chladni method) of the plates were measured contactless with the boundary conditions not too far from those

I i n the real instrument (cf ., Jansson, Molin, & Sundin, 1970).

I As an example, modes no I, no 3, and no 7 are shown i n Fig. 8a.

The black broad l i n e s ("fringes") give t h e iso-amplitude l ines . The photographs of the TV-screen do, however, not give the credit to the pictures as the same everchanging image gives to the eye. Frequencies of the eigenmodes are given i n Table VIII.

6.2. ~ a l c u l a t i o n s on the top plate of spruce 13.5/3.5, locked at the edcres

The same E'E-model as before was used. Here, we started to evaluate the influence of different boundary conditions of the plate, other than those of the free-free edges. It is, however, clear that the different boundary conditions we use here are not those of the real instrument. The boundary conditions were for case:

a) the edge of the plate is locked i n the out-of-plane direction only, along the inner edge of the ribs. The edge is free to move i n the other directions and angles,

b) as case a but locked also in the out-of-plane direction at the - outer edge of a l l s ix blocks of the violin ( four corner blocks, one upper block, and one lower block).

c) as case b but locked also along the total outer edge of the ribs - e l locked in the out-of-plane directions along the ribs, both inner and outer edges).

A more real is t ic approach would be to include the ribs into our FE- model. S t i l l , the results for the three cases above give interesting l i m i t s to the behavior of the plate. Frequencies of the eigenmodes, see Table M, and m o d a l patterns for case a, see Fig. 8'. -

Fig. .8. a) Shapes of modes no 1 , no 3, and no 7 experimentally obtained by the Vibravision with the top plate glued to ribs;

b) FEM-calculated iso-amplitude lines for the top plate 13.5/3.5 boundary condition case a - the modes no 1 to no 8 (thick lines mark nodal Eries) ;

c) FFM-calculated iso-amplitude lines for the back plate 16.5/3.5 boundary condition case a - the modes no 1 - to no 8.

6.3. Back plate, experimental results Frequencies of the e i g d e s , see Table X, and modal patterns, see

Fig. &. Measurements were made as in Section 6.1. It should be noted that the plate is 3.5 mm thick everywhere except at the edges. Here the thickness is 2.5 mm only. F'urthermore, the uniformity of the glue joint along the ribs was somewhat doubtful.

6.4. J?EM-calculations, the back plate of mple 16/3.5, locked at the edges

Frequencies of the eigenmodes, see W l e XI, and the modal shapes for case a, see Fig. 8c. The numerical calculations were made as in - Section 6.2, i.e., as for a plate of constant thickness 3.5 mm.

mte that the order of modes in Tables VIII, XI and XI are taken I

from the FEM-calculations for spruce in Table M. The order of modes in Tables VIII, XI and XI may not follow frequency order.

Table VIII. Measured frequencies in Hz of the top plate glued to the ribs.

Frequencies 574 848 869 1115 1346 1393 1250 1627

Table M. FEM-calculated frequencies in Hz of the top plate with dif- ferent boundary conditions.

Boundary condition

case a) 357 672 719 920 1104 1310 1315 1700

case b) 456 836 818 1044 1307 1359 1472 1847

case c) 530 916 948 1187 1370 1530 15% -

Table X. Measured frequencies in Hz of the back plate glued to the ribs.

Frequencies 611 1012 989 1228 - 1428 1298 1621

Table XI. FEM-calculated frequencies in Hz of the back plate with different boundary conditions.

Boundary condition case a) 369 722 786 990 1141 1417 1376 1709

case b) 476 885 899 1120 1345 1462 1580 -

case c) 559 1005 1036 1306 1445 1694 1695 -

6.5. Cbnclusions, changes in boundary conditions First, it is clear that neither of the three different boundary

conditions used are close to that of the real instrument, mainly since in-plane motions of the plate at the ribs are free. They should be coupled to the in-plane motions of the ribs as well as to the angular deflections of the ribs. It is our hope to continue our calculations with an FE-model including this. The imperfections probably explain why our model even in case c tends to be too "soft" for the lowest modes of vibration, ia., the calN~ated numbers are higher than the experimental ones. At higher modes of vibration, this error in boundary condition is of less or minor importance as the relative frequency change in higher modes due to increased stiffness in boundary conditions is decreasing, i.e., higher modes are less influenced by the boundaries than lower ones. For higher modes of vibration, however, another source of error with the opposite effect is increasing. Our FE-model tends to get too stiff, thus giving too high frequencies for higher modes, i.e., we should increase the number of finite elements. Effects in these direc- tions, i.e., trx, low frequencies for the first mode ard too high for the higher modes, can be seen in Tables VIII to XI.

Note how important different boundary conditions are to the fre- quency of the modes. For mode no 1 of the top plate, the frequency increases from 357 Hz to 530 Hz, i.e., almost 50% and for mode no 7 the corresponding increase is 20% from case a to case c. - -

Boundary condition c is in fact not far from "clamped" (case c allows in-plane motions and small angles at the edge, though), and this condition gives a considerably more stiff construction than that of case

a which often is called "simply supported". Now, since the first mode in - our experiments had a frequency even higher than that of case c, this - implies that the plate should be treated as close to "clamped". It still remains to be investigated, however, the influence of in-plane forces in the plate at the boundaries. In our present FEM-program, we are not able to lock the plate at the edge for in-plane motions and angles, and, therefore, this remains to be tested. We also want to

include the ribs into our FE-model. This may be the best approach to get correct boundary conditions for the violin plate.

It should be noted that modes no 2 and no 3 often are close in frequency and that they sometimes shift order of appearance, that mode no 6 often is hard to find in the experiments and is close to, and sometimes changes order with, mode no 7.

The calculated modal shapes deviate somewhat frm those measured. The antinode of mode no 1 falls in the center of the "waist" for the calculated shapes but below the waist for the measured modes. m e cal- culated higher modal shapes are, in general, little influenced by the different boundary conditions. The calculated shapes differ though from W s e measured.

An earlier study of a violin in different steps of cmstr\action can be said to continue where this paper ends (Jansson & al., 1970). mere- fore, a comparison with the experimental results is interesting. The earlier top and back plates had about the same shape as our present plates but they were only 2.5 mm thick along the edges. As can be expected, all frequencies are slightly lower than our results of today. Shape and order are the same for the top plates (the present back plate experiments were less well controlled) but mode no 6 is probably missing or has changed order with mode no 7. For the earlier top plate, the antinode of the first mode was in the center of the waist and not somewhat below the waist as for the present plate. Berth the earlier and the present back plates had the same antinode in the same position, i.e., somewhat below the waist. Thus such a Mamental property as the position of the antilaode of mode no 5 could differ considerably.

7. Conclusion By the numerical and physical experiments we wanted to answer the

following questions:

I. Fbw do changes in thickness, thickness distribution, arching, and stiffness distribution influence the eigenmodes of the free violin plate?

11. Can "deviating" material properties be "corrected" for by ad- justing thickness and arch height?

111. If so, how should the thickness and arching be chosen with given material properties of the blarJts (i.e., mass and resonance frequencies of the blank)?

N. What kind of boundary condition is appropriate to the plate when glued to the ribs?

Cbmparisons of the numerically calculated and the experimentally measured properties of corresponding violin plates has shown close agreement. This makes our study in Section 5 plausible and there most of the answers to the first question are given.

It was shown that a 10% increase in overall thickness approximately equals a -12% density shift, a +20% in arch height shift, or a +12% shift in all three elastical parameters simultaneously. By looking at modes no 1, no 2 and no 5 more in detail, the following was found. The resonance frequencies of modes no 1 and no 2 were most sensitive to thickness shifts in areas 4 and 3 while that of mode no 5 was most sensitve to thickness shifts in areas 3 and 2 (cf., Table VI.c and Fig. 6). There was a tendency that the overall thickness influenced the resonance frequencies of modes no 1 and no 2 more than that of mode no 5 and that the arch height influenced the resonance frequency of mode no 5 more than those of modes no 1 and no 2 (Table V1.b and V1.d). Some guidelines for tuning are presented in Section 5.

It was found that different elastical parameters affect the modes differently, for instance, mode 110 1 of the free-free plate is mostly dependent on the shear modulus, mode no 2 somewhat more on 5, ard mode no 5 mostly on EL* The higher modes tended though to be equally de- pendent on all three elastical parameters. The nodal lines of mode no 5, and mainly the nodal line in the lower part, were most sensitive to changes. This change, a tendency to form a corner which "spills over the edge" with thinning, seemed to indicate an increasing relatively longi- tudinal stiffness (increased morlulus EL and/or increased arch height, cf., Figs. 7a and b).

The second and third questions are related and the following an- swers were found:

If all three parameters, i.e., EL, E+ and G, were increased by the same amount, their influence could be counteracted by thinning the plate uniformly or by decreasing the arch height, see Section 5. If, lmwever, only one of the parameters were changed, this effected the modes differ- ently. Therefore, we believe that single parameter deviations are best dealt with by thinning the plate unevenly. For instance, mode no 1 will increase in frequency with an increase in the shear modulus. This increase can be counteracted by thinning the plate in the center (area 4 in Fig. 6). The thinning has, however, also the side effect to decrease the frequency of mode no 21 which is little influenced by the shear modulus. Unfortunately, this type of complication seems to be general - different modes cannot be adjusted independently. A major correction must be followed by minor corrections for side effects. A practical example of "design" from material properties is given in Section 5.6.

?b answer question no 4 we have studied free-free plates as well as a model of the plate when only out-of-plane motions at the boundaries (the edges) are restrained; all other motions at the boundaries (edges) are permitted. This is by no means a perfect model for the conditions in the real violin. Our experiments indicated that the plate slmuld be

treated as approximately clamped. It is, however, evident that a better understanding of the boundary condition sbuld be foLlnd. Dif- ferences in boundary conditions may have larger influence than tkw>se of material parameters. Thus, we plan to include the ribs in our model as well as the glue joint in itself.

The presented work is the result of cooperation between three institutions. The numerical experiments were made at Lul& University of Technology, the physical experiments at the Department of Speech Cbm-

munication and Music Acoustics supplemented with the Vibravision measu- rements at the Institute of Optical Ibsearch, both at the b y a l Insti- tute of Technology, Stockholm. The project was supported by the Swedish Board for Technical Development and the Swedish Natural Science Research Cbuncil .

References FJc, L. & Jansson,E. (1984): *'Vibravision and electroacoustical methods applied to determine vibration properties of wooden 'blanks" for violin plates", SrIGQPSR 4/1984, pp. 39-57.

Hutchins, C. (1980): "Tuning of violin plates", pp. 165-181 in Sourd Generation in Winds, Strings, Computers, byal Swedish Academy of MX Stockholm.

Hutchins, C. (1985): "What the violin maker wants to know'*, pp. 47-67 in SM?C 83, Vol. I1 (A. Askenfelt, S. Felicetti, E. Janssan, & J. Sund- berg, eds.), Publ. issued by the Fbyal Swedish Academy of Music No. 46:2, 1985.

Jansson, E., Molin, N-E., & Sundin, H. (1970): "Resonances of a violin body studied by hologram interferometry and acoustical methods", Physica Scripta - 2, pp. 243-256.

Mattsson, G. (1984) : Personal cornmullciation.

McIntyre, M.E. & Woodhouse, J. (1984): "(21 measuring wood properties, Part I", Catgut Arx>ust.Soc. Newsletter no 42.

McIntyre, M.E. & Woodhouse, J. (1985): "0-1 measuring wood properties, Part 11", Catgut Acoust.Soc. Newsletter no 43.

Molin, N-E., Tinnsten, M a , Wiklund, U., & Jansson, E.V. (1984): "FEM- analysis of an orthotropic shell to determine material parameters of wood and vibration modes of violin plates", SIL-QPSR 4/1984, pp. 11-37.

-el, 0. (1967): Die Kunst des Geigenbaues, third edition, (ed. b~ F Winckel ) , B.F. Voigt Verlag, Hamburg.